What is Binomial: Definition and 667 Discussions

In probability theory and statistics, the binomial distribution with parameters n and p is the discrete probability distribution of the number of successes in a sequence of n independent experiments, each asking a yes–no question, and each with its own Boolean-valued outcome: success (with probability p) or failure (with probability q = 1 − p). A single success/failure experiment is also called a Bernoulli trial or Bernoulli experiment, and a sequence of outcomes is called a Bernoulli process; for a single trial, i.e., n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.
The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacement from a population of size N. If the sampling is carried out without replacement, the draws are not independent and so the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution remains a good approximation, and is widely used.

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  1. Mogarrr

    Parameter space for the negative binomial distribution

    Homework Statement For the negative binomial distribution, with r known, describe the natural parameter space Homework Equations the pmf for the negative binomial distribution with parameters r and p can be 1) P(X=x|r,p)= \binom {x-1}{r-1}p^{r}(1-p)^{x-r} where x=r,r+1,... , or 2)...
  2. Mogarrr

    Negative binomial transformation and mgf

    Imo, this problem is crazy hard. Homework Statement Let X have the negative binomial distribution with pmf: f_X(x) = \binom{r+x-1}{x}p^{r}(1-p)^{x}, x=0.1.2..., where 0<p<1 and r is a positive integer. (a) Calculate the mgf (moment generating function) of X. (b) Define a new...
  3. Greg Bernhardt

    What is the Binomial Theorem and How is it Proven?

    Definition/Summary The binomial theorem gives the expansion of a binomial (x+y)^n as a summation of terms. The binomial theorem for positive integral values of 'n', is closely related to Pascal's triangle. Equations The theorem states, for any n \; \epsilon \; \mathbb{N} (x+y)^n =...
  4. A

    Binomial expansion, general coefficient

    Homework Statement Find the coefficient of x^n in the expansion of each of the following functions as a series of ascending powers of x. \frac{1}{(1+2x)(3-x)} Homework Equations The Attempt at a Solution (1+2x)^{-1} = 1 + (-1)2x + \frac{(-1)(-2)}{2!}(2x)^2 +...
  5. anemone

    MHB Optimizing Binomial Coefficients for Maximum Value

    From the binomial theorem, we have $\displaystyle \begin{align*}\left(1+\dfrac{1}{5}\right)^{1000}&={1000 \choose 0}\left(\dfrac{1}{5}\right)^{0}+{1000 \choose 1}\left(\dfrac{1}{5}\right)^{1}+{1000 \choose 2}\left(\dfrac{1}{5}\right)^{2}+\cdots+{1000 \choose...
  6. A

    Binomial expansion of (1+(1/x))^(-1)

    Expand the following functions as a series of ascending powers of x up to and including the term x^3. In each case give the range of values of x for which the expansion is valid. (1+(1/x))^(-1) The Attempt at a Solution 1 + (-1)(1/x) + (-1)(-2)(1/x^2)/2 + (-1)(-2)(-3)(1/x^3)/3! = 1 -...
  7. E

    Self review: Statistics - Binomial Distribution

    Homework Statement The Binomial Distribution - already developed by Jacob Bernoulli (in 1713), et alii, before Abraham de Moivre (1667-1754 CE), et alii, developed the Normal Distribution as an approximation for it (id est, the Binomial Distribution) - gives the discrete probability...
  8. A

    Coefficient of x^r in Expansion of (1+x)(1-x)^n

    I am puzzled by the following example of the application of binomial expansion from Bostock and Chandler's book Pure Mathematics: If n is a positive integer find the coefficient of xr in the expansion of (1+x)(1-x)n as a series of ascending powers of x. (1+x)(1-x)^{n} \equiv (1-x)^{n} +...
  9. J

    Calculating n!/(k-1)!(n-k+1)! from Binomial Theorem

    1. How do you get n!/(k-1)!(n-k+1)! from \begin{pmatrix} n\\k-1 \end{pmatrix} I thought it would be n!/(k-1)!(n-k-1)! where the n-k+1 on the bottom of the fraction would be a n-k-1 instead. I don't understand why there is a "+1" wouldn't you just replace k with k-1 in the binomial formula?
  10. K

    Why Do I Need to Multiply Probabilities in a Binomial Distribution?

    please refer to the second line of solution, since we only concerned about the probability of getting number (5) , then why can't I just just say P=(5/6)^5 , why should I times =(5/6)^5 with (1/6)^2 ?
  11. DreamWeaver

    MHB Finite Binomial Sum: Proving 1 + 1/2 + 1/3 + ... + 1/n

    Show that \sum_{j=1}^{j=n}\binom{n}{j} \frac{(-1)^{j+1}}{j} = 1 +\frac{1}{2} +\frac{1}{3} + \cdots +\frac{1}{n}
  12. A

    Exploring the Depth of the Binomial Theorem: A Scientist's Perspective

    Hello all! This isn't a problem in particular I'm having trouble with, but a much more general question about the binomial theorem. I'm using Stewart's precal book. The section devoted to the theorem has several problems dealing with proving different aspects of it, mostly having to do with...
  13. 22990atinesh

    Importance of Binomial Theorem

    I know Binomial Theorem is a quick way of expanding a Binomial Expression that has been raised to some power i.e ##(a+b)^n##. But why is it so important to expand ##(a+b)^n##. What is the practical use of this in Science and Engineering.
  14. Saitama

    MHB Evaluating a sum involving binomial coefficients

    Problem: Evaluate $$\mathop{\sum \sum}_{0\leq i<j\leq n} (-1)^{i-j+1}{n\choose i}{n\choose j}$$ Attempt: I wrote the sum as: $$\sum_{j=1}^{n} \sum_{i=0}^{j-1} (-1)^{i-j+1}{n\choose i}{n\choose j}$$ I am not sure how to proceed from here. I tried writing down a few terms but that doesn't seem...
  15. MarkFL

    MHB Legend Of ~Incredim's question at Yahoo Answers regarding binomial probability

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  16. T

    Proving (1+x)g'(x) = kg(x) using Binomial Series | Homework Solution

    Homework Statement g(x) = \sum_{n=0}^\infty \binom{k}{n} x^n g'(x) = k\sum_{n=0}^\infty \binom{k-1}{n} x^n prove that (1+x)g'(x) = kg(x) The Attempt at a Solution k(1+x)\sum_{n=0}^\infty \binom{k-1}{n} x^n distribute k[\sum_{n=0}^\infty \binom{k-1}{n} x^n +...
  17. S

    Binomial series - Finding square root of number problem

    Homework Statement Expand ##(1+x)^(1/3)## in ascending powers of x as far as the term ##x^3##, simplifying the terms as much as possible. By substituting 0.08 for x in your result, obtain an approximate value of the cube root of 5, giving your answer to four places of decimals. Homework...
  18. S

    Need help understanding the binomial series

    Homework Statement My math textbook is currently on the Binomial Series now, after completing the Binomial Theorem (no problems with that one). I believe most of my trouble comes from the book's rather glancing explanation of it, only giving examples of the form ##(1 +/- kx)##. Now have...
  19. E

    Stat mech and binomial distribution

    Homework Statement Suppose that particles of two different species, A and B, can be chosen with probability p_A and p_B, respectively. What would be the probability p(N_A;N) that N_A out of N particles are of type A? The Attempt at a Solution I figured this would correspond to a binomial...
  20. anemone

    MHB Evaluating an Infinite Sum of Binomial Coefficients

    Evaluate $\displaystyle\lower0.5ex{\mathop{\large \sum}_{n=2009}^{\infty}} \dfrac{1}{n \choose 2009}$.
  21. D

    Binomial Distribution: Finding the number of trials

    Homework Statement Question: Find the number of trials needed to be 90% sure of at least three or more success, given that probability of one success is 0.2 Homework Equations N/A The Attempt at a Solution My initial attempt at the problem was finding the probability of at least...
  22. A

    Binomial Distribution: Average & Probability of ≥1 Success

    The average if the binomial distribution with probability k for succes is simply: <> = Nk So this means that if <> = 1 the distribution function must be peaked around 1. In general when is it a good approximation (i.e. when is the function peaked sufficiently narrow) to say that the...
  23. MarkFL

    MHB George's question at Yahoo Answers regarding the binomial theorem

    Here is the question: I have posted a link there to this thread to the OP can view my work.
  24. U

    Summing up binomial coefficients

    Homework Statement The value of ((^n C_0+^nC_3+...) - \frac{1}{2} (^nC_1+^nC_2+^nC_4+^nC_5+...))^2 + \frac{3}{4} (^nC_1-^nC_2+^nC_4-^nC_5...)^2 The Attempt at a Solution I can see that in the left parenthesis, the first bracket contains terms which are multiples of 3 and in the second...
  25. A

    Is binomial distribution approriate

    Suppose I have 6 die and toss them. The probability to have n 6's is binomially distributed with parameter 1/6. Now suppose instead tossing the 6 die and having 1/6 probability for a 6 each dice's probability to show 6 grows continously in the time interval t=0 to t from 0 to 1/6. Can I then...
  26. B

    How do I expand (1 + x)^{2}(1 - 5x)^{14} as a series of powers of x?

    Hello, I have a problem regarding the binomial theorem and a number of questions about what I can and can't do. Homework Statement Write the binomial expansion of (1 + x)^{2}(1 - 5x)^{14} as a series of powers of x as far as the term in x^{2} Homework Equations The Attempt at a Solution I...
  27. MarkFL

    MHB Vandomo's question at Yahoo Answers regarding the binomial theorem

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  28. P

    MHB Expected Value of Negative Binomial Random Variable

    Given $X$ as a negative binomial random variable with parameters $r$ and $p$. Find $E(\frac{r-1}{X-1})$. As $E(g(X))$ is defined as $\sum_{x\in X(\Omega)}g(x)p(x)$, this is my attempt in which I am stuck. What can I do next? In the case $y=r-1$, is the sum invalid? Thanks in advance!
  29. P

    Binomial Central Limit Theorem

    Homework Statement Here are the problems: A roulette wheel has 38 slots, numbered 0, 00, and 1 through 36. If you bet 1 on a specified number, you either win 35 if the roulette ball lands on that number or lose 1 if it does not. If you continually make such bets, approximate the...
  30. T

    Proof by Induction involving Binomial Coefficients

    Homework Statement Prove by induction that for any positive integers a, b, and n, (a choose 0)(b choose n) + (a choose 1)(b choose n-1) + ... + (a choose n)(b choose 0) = (a+b choose n) Homework Equations (x choose y) = (x!)/((x-y)!y!) The Attempt at a Solution I am able to do the...
  31. P

    MHB Simplify equation using binomial theorem

    I'm sure this is easy but it has got me baffled. I'm told that the binomial theorem can be used to simplify the following formula x = \dfrac{1 - ay/2}{\sqrt{1-ay}} to (approximately) x = 1 + a^2 y^2 / 8 if a << 1. Thanks for any help or pointers on this one in particular, and/or general...
  32. C

    Hypothesis test (binomial) problem

    Homework Statement Hester suspected that a die was biased in favour of a four occurring. She decided to carry out a hypothesis test. When she threw the die 15 times, she obtained a four on 6 occasions. Carry out the test, at the 5% level, stating your conclusion clearly. Homework...
  33. pellman

    Expectation value for first success in a binomial distribution?

    This is not a homework problem. Just a curiosity. But my statistics is way rusty. Suppose a binomial probability distribution with probability p for a success. What is the expected number of trials one would have to make to get your first success? In practice, this means if we took a large...
  34. DocZaius

    Poisson vs Binomial approaches yield different results

    Homework Statement I made this question for myself to try to see if I could use two approaches (Poisson Distribution and Binomial Distribution) to solve a problem: Someone's average is to make 1 out of every 3 basketball shots. What are the chances she makes exactly 2 shots in a trial of 3...
  35. S

    What are the constants in the binomial series expansion for (1+mx)^-n?

    Homework Statement For ##n>0##, the expansion of ##(1+mx)^{-n}## in ascending powers of ##x## is ##1+8x+48x^{2}+...## (a) Find the constants ##m## and ##n## (b) Show that the coefficient of ##x^{400}## is in the form of ##a(4)^{k}##, where ##a## and ##k## are real constants. Homework...
  36. P

    Sally's Goal-Shooting - Binomial Distribution Q&A

    Hello all, I just have a question which covers binomial distribution. Sally is a goal shooter. Assume each attempt at scoring a goal is independent, in the long term her scoring rate has been shown as 80% (i.e. 80% success rate). Question: What's the probability, (correct to 3...
  37. D

    Negative binomial distribution

    Homework Statement Repeatdly roll a fair die until the outcome 3 has accurred on the 4th roll. Let X be the number of times needed in order to achieve this goal. Find E(X) and Var(X) Homework Equations The Attempt at a Solution I am having trouble deciphering this question...
  38. MarkFL

    MHB Jeffrrey's question at Yahoo Answers regarding binomial expansion

    Here is the question: I have posted a link there to this topic so the OP can view my work.
  39. srfriggen

    Expand (1-2i)^10 without the Binomial Expansion Theorem

    Homework Statement Expand (1-2i)^10 without the Binomial Expansion Theorem I know I need to put this in polar form and then it's simple from there, however, I am simply having a difficult time finding the angle. Drawing the complex number as a vector in the complex plane I get a...
  40. G

    Can (x+y)^(1/2) be expanded using the binomial series?

    Is it possible to do a binomial expansion of (x+y)^{1/2}? I tried to compute it with the factorial expression for the binomial coefficients, but the second term already has n=1/2 and k=1, which makes the calculation for the binomial coefficient (n 1) weird, I think. Any advice?
  41. I

    Taylor Series, Binomial Series, Third Order Optics

    Homework Statement Show that if cosΦ is replaced by its third-degree Taylor polynomial in Equation 2, then Equation 1 becomes Equation 4 for third-order optics. [Hint: Use the first two terms in the binomial series for ℓ^{-1}_o and ℓ^{-1}_i. Also, use Φ ≈ sinΦ.] Homework Equations Sorry that...
  42. S

    MHB Infinite Sums Involving cube of Central Binomial Coefficient

    Show that $$ \begin{align*} \sum_{n=0}^\infty \binom{2n}{n}^3 \frac{(-1)^n}{4^{3n}} &= \frac{\Gamma\left(\frac{1}{8}\right)^2\Gamma\left(\frac{3}{8}\right)^2}{2^{7/2}\pi^3} \tag{1}\\ \sum_{n=0}^\infty \binom{2n}{n}^3 \frac{1}{4^{3n}}&= \frac{\pi}{\Gamma \left(\frac{3}{4}\right)^4}\tag{2}...
  43. J

    Quotient rule and binomial theorem

    If it's possible to relate the product rule with the binomial theorem, so: (x+y)^2=1x^2y^0+2x^1y^1+1x^0y^2 D^2(fg)=1f^{(2)}g^{(0)}+2f^{(1)}g^{(1)}+1f^{(0)}g^{(2)} So, is it possible to relate the quotient rule with the binomial theorem too?
  44. G

    Fisher's Approximation of a Binomial Distribution

    Homework Statement Suppose that X is the number of successes in a Binomial experiment with n trials and probability of success θ/(1+θ), where 0 ≤ θ < ∞. (a) Find the MLE of θ. (b) Use Fisher’s Theorem to find the approximate distribution of the MLE when n is large. Homework Equations...
  45. Saitama

    MHB Finding b_n - Binomial theorem problem

    Question: If $\displaystyle \sum_{r=0}^{2n} a_r(x-2)^r=\sum_{r=0}^{2n} b_r(x-3)^r$ and $a_k=1$ for all $k \geq n$, then show that $b_n={}^{2n+1}C_{n+1}$. Attempt: I haven't been able to make any useful attempt on this one. I could rewrite it to: $$\sum_{r=0}^{n-1} a_r(x-2)^r +...
  46. MarkFL

    MHB Macie's question at Yahoo Answers regarding binomial probability

    Here is the question: I have posted a link there to this topic so the OP can see my work.
  47. N

    MHB Binomial Problem: I Don't Understand i=6

    please refer to attached image I don't understand the reasoning for i=6 since a qualified candidate must have answered 15+ questions correctly, we have i being summed from 15 to 20 for the first part. shouldn't the second part be summed from i=15 to 20? for example, $(20,6)$ would imply, 6...
  48. anemone

    MHB How can we approach a seemingly senseless binomial expansion problem?

    Hi MHB, I've come across this problem and I think I've observed a pattern when I tried to solve it by using the method of comparison with some lower values of the exponents, but then I just couldn't deduce the answer to the problem because the pattern suggests that I can't. Here is the...
  49. R

    MHB Divisibility of Binomial Coefficients

    Hi all, I am trying to figure out if there is a pre-existing theorem and proof of whether or not each of the binomial coefficients in a binomial expansion of (a +b)^n are divisible by n, particularly in the case where n is a prime number. Has this already been asked and answered somewhere in...
  50. P

    Binomial theorem proof by induction

    On my problem sheet I got asked to prove: ## (1+x)^n = \displaystyle\sum _{k=0} ^n \binom{n}{k} x^k ## here is my attempt by induction... n = 0 LHS## (1+x)^0 = 1 ## RHS:## \displaystyle \sum_{k=0} ^0 \binom{0}{k} x^k = \binom{0}{0}x^0 = 1\times 1 = 1 ## LHS = RHS hence true for...
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